# How to get ionic radii for coordination number 12?

I am studying $$\ce{ABX3}$$ perovskites, and I would like to calculate Goldschmidt tolerance factors for them. The $$\ce{A}$$ sites in these materials have a coordination number $$12$$.

The Shannon's paper (R. D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta. Cryst. A 32, 751 (1976)) provides values for ionic radii for the coordination number $$12$$ only for a few elements, but doesn't do it for most other elements.

Are there any sources, where I can get ionic radii for $$\text{C.N.} = 12$$ for all elements?

• Goldschmit has been shown to be a poor model to use with perovskites. – A.K. Oct 18 '18 at 12:34
• How many ionic compounds give coordination number 12? Close-packed structures forcing like charges together don't work well for ionic compounds. Our experimental database might be a bit limited. – Oscar Lanzi Oct 18 '18 at 12:36

The commonly used method of obtaining ionic radii for higher coordination numbers (C.N.) is to extrapolate values from the Shannon's scale  using the relationship between ionic radius and coordination number proposed by Zachariasen :

... the bond lengths $$D(N_1)$$ and $$D(N_2)$$ for cation coordination numbers $$N_1$$ and $$N_2$$ were related as follows

$$D(N_2)=D(N_1)(N_2A_1/N_1A_2)^{1/n} \tag{1}$$

where $$n + 1$$ is the exponent in the Born repulsion term of the lattice energy and $$A_1/A_2$$ is the ratio of the Madelung constants. The quantity $$A_1/A_2$$ was taken to be $$0.972$$ for $$N_2/N_1 = 12/9 = 8/6 = 4/3$$ and $$0.927$$ for $$N_2/N_1 = 12/8 = 9/6 = 6/4 = 3/2$$.

Jia  used Zachariasen's formula and published effective ionic radii for all lanthanides with C.N. $$6$$ to $$12$$.

Using the same approach, selected ionic radii for the 12-coordinated $$\ce{A}$$-sites in oxo-perovskites $$\ce{ABO3}$$  as well as halide perovskites $$\ce{ABX3}$$  have been determined.

### References

1. Shannon, R. D. Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides. Acta Cryst. A 1976, 32 (5), 751–767. https://doi.org/10.1107/S0567739476001551.
2. Zachariasen, W. H. Bond Lengths in Oxygen and Halogen Compounds of d and f Elements. Journal of the Less Common Metals 1978, 62, 1–7. https://doi.org/10.1016/0022-5088(78)90010-3.
3. Jia, Y. Q. Crystal Radii and Effective Ionic Radii of the Rare Earth Ions. Journal of Solid State Chemistry 1991, 95 (1), 184–187. https://doi.org/10.1016/0022-4596(91)90388-X.
4. Li, C.; Soh, K. C. K.; Wu, P. Formability of $$\ce{ABO3}$$ Perovskites. Journal of Alloys and Compounds 2004, 372 (1), 40–48. https://doi.org/10.1016/j.jallcom.2003.10.017.
5. Li, C.; Lu, X.; Ding, W.; Feng, L.; Gao, Y.; Guo, Z. Formability of $$\ce{ABX3}$$ ($$\ce{X = F, Cl, Br, I}$$) Halide Perovskites. Acta Cryst. B 2008, 64 (6), 702–707. https://doi.org/10.1107/S0108768108032734.